Year: 2026
Type: preprint
Abstract: We study the decomposed Möbius energies $E_1$ and $E_2$ at the round circle and determine their complete normal Hessians: $H_1(C)=2π(|D_s|^3-|D_s|)I_2$ and $H_2(C)=-(4π/3)(|D_s|^3-|D_s|)I_2$. Thus $H_1(C)=3H_E(C)$ and $H_2(C)=-2H_E(C)$, where $H_E(C)$ is the Hessian of the full Möbius energy. This exact proportionality singles out the Möbius-invariant combination $F=2E_1+3E_2+2π^2$. We prove that $F(C)=0$ and that its first three derivatives vanish on the full space of parametrized variations. For scalar binormal variations, the fourth variation has a Fourier tensor with complete $3+1$ resonance cancellation and finite $2+2$ fixed-sum blocks. Their positivity follows from an explicit weighted-difference tridiagonalization and a closed determinant formula. For arbitrary normal variations in $\mathbb{R}^3$, Möbius invariance in $\mathbb{R}^4$ reduces the radial component to a second binormal direction. The resulting two-component quartic form splits into trace, symmetric-traceless, and antisymmetric blocks, which are positive on the active coordinates. Consequently, $D^4F(C)[u,u,u,u]\geq 0$ for every smooth real normal field $u$, with equality exactly on the normal tangent space of the Möbius family of round circles. This yields a fourth-order nondegeneracy phenomenon that is not determined by the total-energy Hessian alone.
Source: arXiv (Cornell University)
Author: Ishizeki, Aya
Institution: Saitama University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220381477